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The Beltrami Equation
 
Tadeusz Iwaniec Syracuse University, Syracuse, NY
Gaven Martin Massey University, Albany, Auckland, New Zealand
The Beltrami Equation
eBook ISBN:  978-1-4704-0499-4
Product Code:  MEMO/191/893.E
List Price: $68.00
MAA Member Price: $61.20
AMS Member Price: $40.80
The Beltrami Equation
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The Beltrami Equation
Tadeusz Iwaniec Syracuse University, Syracuse, NY
Gaven Martin Massey University, Albany, Auckland, New Zealand
eBook ISBN:  978-1-4704-0499-4
Product Code:  MEMO/191/893.E
List Price: $68.00
MAA Member Price: $61.20
AMS Member Price: $40.80
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1912008; 92 pp
    MSC: Primary 35;

    The “measurable Riemann Mapping Theorem” (or the existence theorem for quasiconformal mappings) has found a central rôle in a diverse variety of areas such as holomorphic dynamics, Teichmüller theory, low dimensional topology and geometry, and the planar theory of PDEs. Anticipating the needs of future researchers, the authors give an account of the “state of the art” as it pertains to this theorem, that is, to the existence and uniqueness theory of the planar Beltrami equation, and various properties of the solutions to this equation. The classical theory concerns itself with the uniformly elliptic case (quasiconformal mappings). Here the authors develop the theory in the more general framework of mappings of finite distortion and the associated degenerate elliptic equations.

    The authors recount aspects of this classical theory for the uninitiated, and then develop the more general theory. Much of this is either new at the time of writing, or provides a new approach and new insights into the theory. Indeed, it is the substantial recent advances in non-linear harmonic analysis, Sobolev theory and geometric function theory that motivated their approach here. The concept of a principal solution and its fundamental role in understanding the natural domain of definition of a given Beltrami operator is emphasized in their investigations. The authors believe their results shed considerable new light on the theory of planar quasiconformal mappings and have the potential for wide applications, some of which they discuss.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Quasiconformal mappings
    • 3. Partial differential equations
    • 4. Mappings of finite distortion
    • 5. Hardy Spaces and BMO
    • 6. The principal solution
    • 7. Solutions for integrable distortion
    • 8. Some technical results
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1912008; 92 pp
MSC: Primary 35;

The “measurable Riemann Mapping Theorem” (or the existence theorem for quasiconformal mappings) has found a central rôle in a diverse variety of areas such as holomorphic dynamics, Teichmüller theory, low dimensional topology and geometry, and the planar theory of PDEs. Anticipating the needs of future researchers, the authors give an account of the “state of the art” as it pertains to this theorem, that is, to the existence and uniqueness theory of the planar Beltrami equation, and various properties of the solutions to this equation. The classical theory concerns itself with the uniformly elliptic case (quasiconformal mappings). Here the authors develop the theory in the more general framework of mappings of finite distortion and the associated degenerate elliptic equations.

The authors recount aspects of this classical theory for the uninitiated, and then develop the more general theory. Much of this is either new at the time of writing, or provides a new approach and new insights into the theory. Indeed, it is the substantial recent advances in non-linear harmonic analysis, Sobolev theory and geometric function theory that motivated their approach here. The concept of a principal solution and its fundamental role in understanding the natural domain of definition of a given Beltrami operator is emphasized in their investigations. The authors believe their results shed considerable new light on the theory of planar quasiconformal mappings and have the potential for wide applications, some of which they discuss.

  • Chapters
  • 1. Introduction
  • 2. Quasiconformal mappings
  • 3. Partial differential equations
  • 4. Mappings of finite distortion
  • 5. Hardy Spaces and BMO
  • 6. The principal solution
  • 7. Solutions for integrable distortion
  • 8. Some technical results
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.